The Magnetic Part of the Weyl Tensor, and the Expansion of Discrete Universes

Transcription

1 The Magnetic Part of the Weyl Tensor, and the Expansion of Discrete Universes Timothy Clifton Daniele Gregoris Kjell Rosquist Abstract We examine the effect that the magnetic part of the Weyl tensor has on the large-scale expansion of space. This is done within the context of a class of cosmological models that contain regularly arranged discrete masses, rather than a continuous perfect fluid. The natural set of geodesic curves that one should use to consider the cosmological expansion of these models requires the existence of a non-zero magnetic part of the Weyl tensor. We include this object in the evolution equations of these models by performing a Taylor series expansion about a hypersurface where it initially vanishes. At the same cosmological time, measured as a fraction of the age of the universe, we find that the influence of the magnetic part of the Weyl tensor increases as the number of masses in the universe is increased. We also find that the influence of the magnetic part of the Weyl tensor increases with time, relative to the leading-order electric part, so that its contribution to the scale of the universe can reach values of 1%, before the Taylor series approximation starts to break down. T. Clifton School of Physics and Astronomy, Queen Mary University of London, UK. t.clifton@qmul.ac.uk D. Gregoris Department of Mathematics and Statistics, Dalhousie University, Canada. K. Rosquist Department of Physics, Stockholm University, Stockholm, Sweden.

2 2 Timothy Clifton et al. 1 Introduction The large-scale expansion of the Universe is usually taken to be dominated by the Newtonian part of the gravitational field. This idea probably originates from the close correspondence between the Friedmann equations of general relativity and the equations that govern Newtonian cosmologies [1], but also has strong support from rigourous constructions that are based on perturbative expansions of Einstein s field equations [2, 3]. Nevertheless, there are good reasons to be interested in relativistic effects in cosmology. These include the apparent existence of Dark Energy, as well as the dawn of the age of precision cosmology. Put bluntly, relativistic effects need to be understood in order to have faith in the cosmological models that we use to interpret observational data. Geometrically, the electric part of the Weyl tensor is sufficient to determine the Newtonian part of the free gravitational field. The magnetic part of the Weyl tensor, on the other hand, describes other aspects of the relativistic gravitational field [4] (both of these objects are defined in Section 3, below). Well known relativistic effects such as frame-dragging and gravitational radiation require a non-vanishing magnetic Weyl tensor in order to exist, but the effects of the magnetic part of the Weyl tensor on the large-scale expansion of the Universe are still largely unknown. This is partly due to an absence of realistic cosmological models that have non-zero magnetic Weyl curvature, and that could therefore be considered to be non-silent [5]. This situation is, however, changing. It has recently been demonstrated that cosmological models that contain regularly-arranged discrete masses can generate a non-zero magnetic Weyl tensor, even if no such curvature existed in their initial data [6]. The authors of this study considered the effect that this tensor has on the large-scale expansion of a universe that contains eight black holes, using both the leading-order term in a Taylor series expansion, and by numerically integrating the evolution equations of the initially silent space. We extend their study by calculating both the leading-order and next-to-leading-order parts of the relevant series expansion, and by using these results to determine the effect of the magnetic part of the Weyl tensor on the large-scale expansion of all cosmological models that contain regularly arranged discrete masses in a closed cosmology. The phenomenology of such models is interesting, as they can be considered a first approximation to the type of universe within which we actually live. More mathematically, they provide a tractable way to formulate n-body cosmology as a relativistic initial value problem. They therefore provide an ideal arena within which to study relativistic gravitational effects in cosmology. Of course, the magnetic part of the Weyl tensor is a frame-dependent object, and its magnitude will change when different sets of observers are considered. However, in cosmology it is natural to consider sets of time-like curves that are both geodesic, and (in some sense) comoving with the objects that exist within the space-time. It is with respect to observers following such a set of curves that one can talk about cosmological expansion, and it is with

3 The Magnetic Part of the Weyl Tensor, and the Expansion of Discrete Universes 3 respect to the same set of curves that we will talk about the magnetic part of the Weyl tensor. The influence of the magnetic part of the Weyl tensor on the cosmological expansion is of special interest because virtually all cosmological solutions of Einstein s equations are derived under assumptions that force it to be zero (for cosmologically interesting congruences of curves). Such situations are, however, highly unrealistic, as observers that follow the geodesic set of time-like curves that describe the world-lines of real galaxies will certainly experience the consequences of this part of the curvature of space-time, which is in general non-zero. We must therefore precisely quantify the effects of the magnetic part of the Weyl tensor on the large-scale expansion of space, if we are to fully understand the recessional velocity of galaxies in the real Universe (and all of its associated consequences). In Section 2 we describe in detail the cosmological models that we will be using in this study. They consist of regularly-arranged discrete masses in a closed space, and are formulated as a relativistic initial value problem. In Section 3 we then introduce the compact formalism that will be used in the rest of the paper. This starts by using a 1+3 decomposition of the geometric and kinematic quantities involved, and finishes with a decomposition that can be efficiently used to exploit the symmetries of the problem. The explicit form of the symmetry restrictions are then determined in Section 4, where it is shown that the influence of the magnetic part of the Weyl tensor on the relevant evolution equations does not necessarily vanish. Section 5 then contains some lengthy calculations to determine the coefficients of a Taylor series expansion that can be used to include the effect of the magnetic Weyl tensor on the large-scale expansion of space. In Section 6 we present the numerical results for each of the lattice universes that we will be considering, before concluding in Section 7. Throughout the paper we use the first half of the Latin alphabet to denote space-time coordinate indices, and the second half to denote spatial coordinate indices. Greek letters are reserved to denote spatial frame indices, where they are required. 2 Lattice Models of the Universe The space-times we wish to consider are those that have come to be known as lattice models, in some parts of the cosmology literature. These models have a periodic structure, and are constructed from a number of regularly arranged cells, each of which are identical to one another. Such models have been studied in a number of different contexts over the past few years, as they offer a simple enough setting to provide concrete answers to questions of direct physical interest. The particular type of lattices we wish to consider are those in which the topology of the cosmological region is a hypersphere. Furthermore, we want to consider situations where the only mass present is in the form of black holes, without angular momentum or electric charge. We position one single black

4 4 Timothy Clifton et al. hole at the centre of every lattice cell, so that the cosmological region as a whole contains a regular array of these objects. The geometry of the spacetime can then be treated as a vacuum solution of Einstein s equations, with its scale and expansion being prescribed accordingly. In the rest of this section we will describe the specifics of the six different lattice configurations that we will be studying, as well as the solutions to the constraint equations, and the existence of curves within the space-time that display special rotational symmetries. This recaps results from previous work [7 9]. 2.1 The Six Closed Lattices Before solving for the geometry of space-time, we need to understand how to divide a hyperspherical universe into n identical cells. If we choose each cell to be one of the five possible Platonic solids (i.e. one of the five possible regular convex polyhedra), then there are six different ways that we can arrange these cells to form a closed lattice. Each of these configurations corresponds directly to one of the six regular, convex polychora that exist in four dimensions. They are [10]: 5-cell. This lattice consists of five tetrahedral cells. Three cells meet along every cell edge, and there are a total of five vertices (the points where cell corners meet). This is the smallest lattice that can be constructed from regular convex polyhedra. 8-cell. This structure is composed of eight cubes, arranged so that three cells meet along every cell edge, giving a total of sixteen vertices. It corresponds to the space-time studied numerically in [6, 11]. 16-cell. This lattice is made from sixteen tetrahedral cells, with four cells meeting at every cell edge. It is dual to the 8-cell (meaning that interchanging vertices and cell centres gives an 8-cell lattice). 24-cell. The 24-cell is constructed from twenty four octahedral cells. Three of these cells meet along every cell edge, and there are a total of twenty four vertices. The 24-cell is self-dual, as it has the same number of cells and vertices. 120-cell. This structure contains one hundred and twenty dodecahedral cells, with three cells meeting at every edge. There are six hundred vertices in this structure, meaning that it is dual to the 600-cell, described below. 600-cell. This is the largest of all lattices that fits our specifications. It is built from six hundred tetrahedral cells, with five of these meeting along every edge. In the rest of this section we will outline how Einstein s constraint equations can be solved for these lattices, as well as how curves that exhibit local rotational symmetry can be identified.

5 The Magnetic Part of the Weyl Tensor, and the Expansion of Discrete Universes Initial Data If we treat general relativity as an initial value problem, then we can split Einstein s equations into constraint and evolution equations. The former of these can then be written as a Hamiltonian constraint and a momentum contraint, such that, in vacuum, (3) R + K 2 K ij K ij = 0 (1) D j K j i D i K = 0, (2) where (3) R is the Ricci scalar of the geometry on the initial hypersurface, Σ, and K ij is its extrinsic curvature in the embedding space-time. The D i derivative is used here to denote a covariant derivative within Σ, and K is the trace of K ij. Eqs. (1) and (2) are, in general, very difficult to solve. However, we can simplify the problem dramatically if we choose our initial hypersurface carefully. For example, if we choose it to be symmetric under time reversal then we automatically have K ij = 0. The Hamiltonian constraint then reduces to (3) R = 0, (3) while the momentum constraint is identically satisfied. This equation is much easier to solve, and is the basis of the study of geometrostatics (the gravitational analogue of the field of electrostatics [12]). We will assume initial data of exactly this type in what follows, and find solutions to the constraint equation (3) that describe the maximum of expansion of our lattices. These results were presented for the first time in [7]. A metric ansatz for the initial geometry can be chosen as follows: dl 2 = ψ 4 ( dχ 2 + sin 2 χdω 2), (4) where dω 2 = dθ 2 + sin 2 θdφ 2 is the line-element of a 2-sphere, and where ψ = ψ(χ, θ, φ) acts as the square root of the scale factor. The time-symmetric Hamiltonian constraint then reduces to 2 ψ = 3 4 ψ, (5) where 2 is the Laplacian operator on the conformal 3-sphere. This equation is now linear in ψ, meaning that if a solution can be found for a single black hole, then we can simply super-impose this with other solutions of the same form to find multi-black hole solutions. This is exactly what we need to form our lattice models of the Universe. A simple solution to Eq. (5) is given by ψ = m/(2 sin ( ) χ 2 ), where m is a constant. This function has a pole at χ = 0, and looks very much like the gravitational field one might expect for a point-like particle located at that position. Of course, there is nothing special about the point χ = 0, and we can generate any number of similar functions by rotating the coordinate system so

6 6 Timothy Clifton et al. that {χ, θ, φ} {χ i, θ i, φ i }, and then replacing χ by χ i in the denominator. This gives the solution for N such particles as N mi ψ = 2 sin ( χ i ), (6) 2 i=1 where χ i should be understood as the value of χ in a coordinate system that has been rotated so that the ith particle appears at χ i = 0. This is just what we need to construct our lattice cosmologies, as it is a solution that corresponds to N arbitrarily located black holes, in a universe that is at its maximum of expansion. In the sections that follow, we will use the solution given by Eqs. (4) and (6) as initial data for our cosmology. This initial data is exact, but has some subtleties to its interpetation. The parameter m i that appears in Eq. (6) looks like it corresponds to the mass of a particle positioned at χ i = 0, but it is not the proper mass of the ith black hole. This must be found by looking at the geometry of the space in the limit χ i 0, and comparing this to a timesymmetric slice through the Schwarzschild solution. One then finds that the proper mass of each of the black holes is a function of the location and m i of each of the other black holes. The parameter m i can then be interpreted as an effective mass, which includes information about the gravitational field of the gravitational potential energy, as well as that from the proper mass of each of the individual black holes [7,13]. As we want all of the black holes in our lattice models to have the same mass, we choose m i = m for all values of i. 2.3 Curves with a Rotationally Symmetric Tangent Space There exists a number of curves, within our lattices, that display Local Rotational Symmetry (LRS) [8, 9]. Perhaps the simplest of these curves to visualize is the example of a cell edge. In each of the lattice structures listed above there were between three and five identical cells meeting along every cell edge. This means that the geometry of space-time in the vicinity of these curves must be invariant under a rotation that interchanges the cells. If we pick a set of orthonormal frame vectors for our 3-space {e 1, e 2, e 3 }, and align e 1 with this special curve, then the rotational symmetry implies [8] V 2 = V 3 = 0 = T 23 and T 22 = T 33, (7) where V α and T αβ are the spatial frame components of any vector V, and any symmetric tensor T. Furthermore, the commutators of the basis vectors are restricted by [8] γ 1 21 = γ 1 31 = 0, γ 2 12 = γ 3 13 and γ 3 12 = γ 2 13, (8) where [e α, e β ] = γ ɛ αβ e ɛ. The results in Eqs. (7) and (8) are only valid in the tangent spaces of the points that make up these LRS curves, but they are significant restrictions.

7 The Magnetic Part of the Weyl Tensor, and the Expansion of Discrete Universes 7 As well as cell edges, there are other curves in our lattices that also exhibit rotational symmetry [9]. They are given by the lines that connect the horizon of the black hole at the cell centre with the cell corner, and the line that connects this horizon with the centre of a cell face. Such curves are illustrated in Fig. 1, for the particular case of a cubic cell. They will be the curves that we solve for in Section 6. The curves that connect the centre of the cell with the cell corner, and the centre of a cell face, are formally infinitely long in the initial data, which is why we have chosen to cut them off at the black hole horizon. They then have finite proper length. The location of the horizon will be determined by assuming the horizon to be initially non-expanding. In this case, the location of the horizon along LRS curves is found by looking for solutions to the following equation [8]: D e1 E 11 = 0, (9) where e 1 is again aligned with the direction of the LRS curves, and E 11 is the frame component of the electric part of the Weyl tensor. The location of the horizon at subsequent times is found by propagating null geodesics along the LRS curves, from the initial location (as can be seen to be required by the null Raychaudhuri equation [8]). 3 Covariant Formalism and Field Equations In order to exploit the symmetries of a space-time it is often convenient to decompose the geometric objects that exist within it. This can be achieved by picking out vector fields that are either aligned with, or orthogonal to, certain invariantly defined sub-spaces, and then by defining projection operators associated with these vectors. Within the field of cosmology, the most widely applied decomposition of this type is the decomposition pioneered by Ehlers [14], and developed by Ellis and others (see e.g. [15,16]). If there also exists a preferred space-like direction within the space, as in LRS space-times [17], then the decomposition of Clarkson becomes a useful tool [18]. Going further, and decomposing using a full set of four mutually orthogonal unit vectors, one is then led to the orthonormal frame approach [19]. In this section we will outline the and approaches, as relevant for our study. 3.1 The Decomposition If there exists a time-like unit vector field, u, then we can define tensors that project parallel and othogonal to this vector, respectively, as U a b := u a u b and h a b := δ a b + u a u b, (10)

8 8 Timothy Clifton et al. Fig. 1 An illustration of the curves that connect the centre of the cell with the centre of a cell face (green), and the centre of a cell with a cell corner (blue). Also displayed is a curve that connects the centre of a cell edge with a cell corner (red). These are the three curves whose evolution will be solved for in Section 6. where δ a b indicates the Kronecker delta. We can also project all derivatives along and orthogonal to u a, respectively, by defining Ṡ a...b c...d := u e e S a...b c...d (11) D a S b...c d...e := h f a h b g... h c hh i d... h j e f S g...h i...j, (12) where S a...b c...d is any tensor. It can immediately be seen that ḣab = 2 u (a u b) and D a h bc = 0, which can be considered the evolution and constraint equations for h ab. The covariant derivative of u a can then be decomposed into its irreducible components, such that a u b = u a u b Θh ab + σ ab + ω ab, (13) where u a = u b b u a is the acceleration vector, Θ := u a ;a is the expansion scalar, σ ab := u (a u b) + (a u b) 1 3 Θh ab is the symmetric trace-free shear tensor, and ω ab := u [a u b] + [a u b] is the skew-symmetric vorticity tensor. These

9 The Magnetic Part of the Weyl Tensor, and the Expansion of Discrete Universes 9 last three quantities describe the rate at which the integral curves of u a expand, squash and twist around each other. It is straightforward to verify that U ab σ bc = U ab ω bc = 0 = U ab u b, meaning that all of these quantities are already projected orthogonal to u a. The fundamental object of interest for describing the free gravitational field, and which is required to complete a description of the vacuum scenario, is the Weyl tensor, C abcd. The existence of the time-like vector field u can be used to split the Weyl tensor into electric and magnetic parts, defined respectively by E ab := C acbd u c u d and H ab := 1 2 ɛ acdc cd beu e, (14) where ɛ abc = η abcd u d is the spatial skew symmetric permutation tensor (defined such that η 0123 = g ). The electric part of the Weyl tensor contains all information about the tidal forces due to gravity. The magnetic part, on the other hand, contains all other information about the Weyl curvature. We also have ɛ abc = η abcd u d and D a ɛ abc = 0, which can be thought of as the evolution and constraint equations for ɛ abc. 3.2 The Vacuum Field Equations In order to write the field equations in the most concise form possible, it is convenient to introduce the definitions of the covariant divergence and curl, which when operating on rank-2 tensors are given by (div S) a := D b S ab and curls ab := ɛ cd(a D c S b) d. (15) It is also convenient to introduce the projected, symmetric, trace-free part of a rank-2 tensor by using angular brackets, such that S ab := h a c h b d S (cd) 1 3 S cdh cd h ab. (16) Note that this means σ ab = σ ab, E ab = E ab, and H ab = H ab, as these tensors are already projected, symmetric and tracefree. With the definitions of all of these quantities and operators in hand, it is now possible to write the vacuum evolution equations for a geodesic ( u a = 0), irrotational (ω ab = 0) flow as [20] Θ = 1 3 Θ2 σ ab σ ab (17) σ ab = 2 3 Θσ ab σ, σ ab E ab (18) Ė ab = ΘE ab + 3 σ, E ab + curlh ab (19) Ḣ ab = ΘH ab + 3 σ, H ab curle ab, (20)

10 10 Timothy Clifton et al. with the constraint equations taking the form (div σ) a = 2 3 D aθ (21) curl σ ab = H ab (22) (div E) a = [σ, H] a (23) (div H) a = [σ, E] a. (24) The bracket operators in these equations are defined as [S, T ] a := ɛ abc S b dt cd, and S, T ab := S c a T c b. This extremely compact way of writing the field equations will greatly aid the manipulations we perform in Section 5. It also has the distinct advantage that all quantities involved have a direct physical interpretation, allowing for a better intuitive understanding of the physical set-up. 3.3 The Decomposition Just as in the previous sections, and following Clarkson [18], if there exists a space-like unit vector field, n, then we can decompose all geometric objects into parts that are parallel and orthogonal to this field. The goal here is to first perform the decomposition with respect to u a, and then to take the tensors projected into the space orthogonal to u a and decompose them further with respect to n a. To this end, we introduce the following projection tensors: N a b := n a n b and f a b := h a b n a n b, (25) where h ab is the projection tensor from above. The tensor N a b can operate on any geometric object, and projects it along n a. The tensor f a b, on the other hand, only makes sense as a projection tensor when it operates on objects that have already been projected using h a b. If na is surface forming, then f a b projects such objects onto the 2-dimensional sheets orthogonal to n a. Under these definitions, we can irreducibly decompose any spatially projected vector, V a = h a b V b, into a scalar part V := V a n a, and a transverse vector part V a := f b a V b. The projected vector can then be written as V a = V a + V a (26) where V a = N b a V b. These definitions mean that f a b V a = N a b V a = 0, which shows that V a has no parts in the sheet, and V a has no part parallel to n a. The other geometric objects we wish to decompose with respect to n a are the symmetric trace-free tensors defined above. Any such tensor can be irreducibly decomposed into a scalar S := N ab S ab, a transverse vector S a := f a b n c S bc, and a transverse and tracefree tensor S ab := (f (a c f b) d 1 2 f abf cd )S cd. The full symmetric and trace-free tensor can then be written in terms of these fundamental objects as S ab = S ab + S ab + S ab, (27)

11 The Magnetic Part of the Weyl Tensor, and the Expansion of Discrete Universes 11 where S ab = S(N ab 1 2 f ab), S ab = 2n (a S b), and S ab = (f (a c f b) d 1 2 f abf cd )S cd. These three objects are each symmetric, and trace-free with respect to h ab. They are the only three such objects that can be constructed from a single power of S, S a and S ab, and as such they specify a unique decomposition of S ab. In the next section we will consider the restrictions that reflection symmetry imposes upon these objects, and why this means that H ab can have an effect on the large-scale expansion of space. 4 Reflection Symmetric Surfaces The presence of reflection symmetric surfaces can greatly simplify the formulation of problems in general relativity. Indeed, it is the presence of a reflection symmetry in the time-like direction that allowed the constraint equations to be solved for a lattice universe [7]. Reflection symmetry in the space-like directions normal to certain dimensional surfaces within the space-time also lead to the notion of piecewise silence, as gravitational waves are forbidden from travelling between lattice cells, or the various chambers within cells [9]. In this section we will investigate the properties of reflection symmetric surfaces further, by formulating their consequences for geometric objects in the decomposition, and by considering the situation where multiple reflection symmetric surfaces intersect (as often occurs in our lattice universe construction). 4.1 Polar Tensors and Axial Tensors When determining the consequences of reflection symmetry for tensorial quantities it is important to distinguish between polar and axial tensors. If a tensorial object can be defined without reference to a specific frame, then that object is said to be a polar tensor. By contrast, if such an object is defined with respect to some fixed direction or combination of directions, then that object is said to be an axial tensor. The 3-dimensional Levi-Civita tensor, ɛ abc, is the archetypical axial tensor. If {e 1, e 2, e 3 } constitute a set of orthonormal spatial frame vectors, then the frame components of this tensor are given by ɛ 123 = 1, and all permutations. In other words, the Levi-Civita tensor defines a preferred orientation of the spatial frame. A reflection, or any other improper orthonormal transformation, changes the orientation of the spatial frame. Reflections therefore change the sign of ɛ abc, meaning that it should be properly identified as a pseudotensor, rather than as a tensor. This distinction is extremely important when determining the consequences of reflection symmetry on geometric quantities. To illustrate this in a formal way, we note that an inversion can be represented by the orthonormal transformation matrix I = 1, where 1 denotes the 3-dimensional unit matrix. Applying the standard tensor transformation

12 12 Timothy Clifton et al. formula for an orthogonal transformation matrix O a b would then give O(S abc... ) = O d a O e b O f c... S def.... (28) If we take S abc... = ɛ abc and O = I then this law would give us O(ɛ abc ) = ɛ abc. The sign change indicates that the transformed object, O(ɛ abc ), has opposite parity to the untransformed object, ɛ abc. This is an unwanted feature if we require the parity of the Levi-Civita tensor (right or left-handedness) to be invariant under any orthogonal transformation. The standard way around this is to modify the transformation rule for pseudotensors, so that it becomes [21] O(S abc...) = det(o)o d a O e b O f c... S def.... (29) We can then transform tensors according to Eq. (28), whilst transforming pseudotensors according to the expression in Eq. (29). The extra factor of det(o) in this latter equation means that any quantity that contains an odd number of Levi-Civita tensors picks up an extra minus sign under any improper orthogonal transformation, such as a reflection. 4.2 Implementing reflection symmetries Consider a reflection symmetry in a spatial hypersurface. Such a symmetry has fixed points that form an invariant symmetry surface, S. Let {n, k, l} be an orthonormal frame adapted to the symmetry in such a way that n is orthogonal to S, and consequently that k and l are parallel to S. The reflection symmetry in the tangent space of a point in S can then be represented by the orthonormal frame transformation matrix R a b = diag( 1, 1, 1) for a polar tensor, and by R = R for an axial tensor. The restrictions on a tensor Q, imposed by invariance under a reflection symmetry, is then determined by the equations RQ = Q or RQ = Q, (30) depending on whether Q is a polar or an axial tensor, respectively. We note that, for the specific fields that appear in Eqs. (17)-(24), the set {Θ, σ ab, E ab } are polar, while H ab is axial (as it involves the Levi-Civita pseudotensor). The only operation in the field equations that changes the parity of an object is the curl, which means that curlh ab is polar, while curlσ ab and curle ab are axial tensors. Decomposed tensor parts, such as E ab, always have the same parity as the full tensor from which they are derived. To calculate the action of the reflection R, let us first consider the case of scalars. We note that the only change possible for a scalar is the change of sign that affects pseudoscalars under an improper transformation, according to the pseudoscalar version of Eq. (29). This implies that all pseudoscalars must vanish on S, leading to the restrictions (curl σ) = (curle) = H = 0. To see the effect on vectorial and tensorial objects we can express the transformation

13 The Magnetic Part of the Weyl Tensor, and the Expansion of Discrete Universes 13 matrix in the form R a b = f a b Na b. The action on the vector part of a tensor, S ab, is then given by 1 R( S ab ) = R c ar d b S cd = 2(f c a N c a)(f d b N d b)n e (c f d) f S ef = S ab. (31) Therefore, in order to fulfil the symmetry condition given in Eq. (30), the vectorial part, S a, of a polar tensor, S a, must vanish on S. It then follows that σ a = E a = ( curlh) a = 0 on S. Applying the reflection transformation in an analogous way to the tensor part of S ab gives R( S ab ) = S ab. Hence, the polar tensor part is not affected by the reflection, and is therefore unrestricted. The axial tensor part, on the other hand, is required to vanish, due to the determinant appearing in Eq. (29). This implies that (curle) ab = H ab = 0. In summary, the tensor parts remaining on a reflection-symmetric surface, after the symmetry restrictions imposed by Eq. (30), are Θ, σ, σ ab, (curlσ) a, E, E ab, (curl E) a, H a, (curlh), and (curlh) ab. 4.3 Intersection of Two Symmetry Surfaces We now consider the symmetry restrictions imposed at points where two reflection invariant surfaces meet. Let us call these two surfaces S k and S l, where the subscripts k and l indicate their normals. The intersection of these two surfaces forms a curve, C, and reflection symmetry about each surface ensures that they meet at right angles to each other. In this case we can choose to adapt the frame in such a way that n is parallel to C. The quantities (curl σ), (curle) and H now have a different interpretation, but they are again pseudoscalars. We therefore have that (curl σ) = (curle) = H = 0. To analyze the restrictions on vectorial and tensorial objects, we note that the reflection transformations associated with k and l can be expressed in the forms R k : γ a b = h a b 2k a k b, R l : γ a b = h a b 2l a l b. (32) Applying R k to the vectorial part of the decomposition gives R k ( S ab ) = 2γ c aγ d bn (c f e d) nf S ef = S ab 4n (a k b) k c S c. (33) Applying the symmetry condition from Eq. (30), for a polar tensor, then leads to k a S a = 0 on S k. A similar argument for the reflection symmetry R l gives l a S a = 0 on S l. It follows that S a = 0 on C, as both of its components must vanish. This implies the restrictions σ a = E a = (curlh) a = 0. For the vector part of an axial vector we get R k ( S ab ) = S ab + 4n (a k b) k c S c. (34) 1 As a technical note, it can be remarked that n a, when regarded as a fixed vector, will be an axial vector. This has the effect that S a and S ab have opposite parity, as their definitions include one and two factors of n a, respectively. This issue is irrelevant for the scalars S and S ab as both contain an even number of factors of n a (2 and 0, respectively).

14 14 Timothy Clifton et al. In this case, the symmetry condition from Eq. (30) takes the traceless symmetric form C ab := n (a S b) n (a k b) k c S c = 0. (35) Only the vectorial component of C ab is non-zero, which means the symmetry condition in Eq. (35) then reduces to C a = 1 2 ( S a k a k b S b ) = 0. (36) This tells us that the component of S a in the l-direction vanishes on S k. Similarly, the symmetry R l implies that the component of S a in the k-direction vanishes on S l, and consequently that the vectorial parts of all axial tensors vanish on S. This gives the restrictions ( curl σ) a = ( curle) a = H a = 0. Let us now consider the consequences of the symmetry conditions, from Eq. (30), on the tensorial part of a polar tensor. We find that the reflection operation gives R k ( S ab ) = S ab 4( S cd k c l d )k (a l b), (37) where we have used the relation S ab k b = ( S bc k b k c )k a + ( S bc k b l c )l a. The symmetry condition in Eq. (30) then implies S ab k a l b = 0. The tensorial parts σ ab, E ab and ( curlh) ab are therefore diagonal, with S ab k a k b as their only independent component. As the right-hand side of Eq. (37) is symmetric in k and l, there are no further restrictions implied by the symmetry action of R l. Finally, we consider the action of the reflection symmetries on axial tensors. They give R k ( S ab ) = R l ( S ab ) = S ab + 4( S cd k c l d )k (a l b), (38) so that the symmetry conditions in Eq. (30) then become S ab = 2( S cd k c l d )k (a l b). (39) This is the condition that S ab k a l b is the only nonvanishing independent component of S ab. This restriction applies to ( curl σ) ab, H ab and ( curle) ab. In summary, at the intersection of two reflection symmetric surfaces we have that the only non-vanishing tensor parts are given by Θ, σ, E and (curlh), the single independent component of the diagonals of σ ab, E ab and (curlh) ab, and the single independent off-diagonal components of (curlσ) ab, (curle) ab and H ab 2. 2 This corrects statements made in [8].

15 The Magnetic Part of the Weyl Tensor, and the Expansion of Discrete Universes Intersection of Three or More Symmetry Surfaces When three or more reflection symmetric surfaces meet the restrictions on geometric objects are much stricter. Such situations arise along LRS curves when the symmetry along the intersecting surfaces is threefold or higher, as was studied in some detail in [8]. Here we adapt our frame to the symmetry curve in the same manner as in the twofold symmetry case, considered above. As we know that all the vectorial and tensorial parts of every geometric object must vanish in this case [8], we only need to consider the scalars. As before, the axial scalars must once again vanish, implying that (curl σ) = (curle) = H = 0. It then follows that curl σ ab = curle ab = H ab = 0. The only remaining parts are therefore given by σ, E and (curlh). If we now re-cast Eqs. (17)-(19), in terms of the quantities that remain after symmetry restrictions, then we find Θ = 1 3 Θ2 3 2 ( σ) 2 (40) σ = 1 ( σ 2Θ + 3 ) σ E (41) 3 2 ( Ė = E Θ 3 ) σ + (curlh). (42) 2 These equations can be further simplified by defining the expansion scalars parallel and perpendicular to n a : H := 1 3 Θ + σ and H := 1 3 Θ 1 2 σ, respectively. The evolution equations for these two quantities are then given by Ḣ + (H ) 2 = E (43) Ḣ + (H ) 2 = 1 2 E, (44) while the evolution equation for the source term, E, is given by Ė + 3H E = (curlh). (45) The scalar (curlh) therefore acts as a source for E, which is itself the source of the expansion along S. This completes our study of the restrictions imposed at reflection-symmetric surfaces. In the following sections we will study the evolution of the non-vanishing parts of the covariant scalars, and in particular the effect that (curlh) has on the expansion of LRS curves with threefold, or higher, reflective symmetries. 5 Taylor Series Expansion of Curl(H) We want to solve the evolution equations (43)-(45), with the initial conditions specified in Section 2. In terms of the variables that result from the decomposition, these initial conditions can be specified as Θ = 0 = σ ab = H ab and Θ = 0 = Ḣab (46)

16 16 Timothy Clifton et al. together with the following conditions on E ab : (div E) a = 0 = (curl E) ab. (47) The first set of initial conditions in Eq. (46) all follow immediately from the requirement of time-reversal symmetry in the initial data [8]. The second set, involving time derivatives, follows from Eq. (17), and from the fact that the Cotton-York tensor on the initial hypersurface is given by C ab = Ḣab [19]. The conditions in Eq. (47) then follow from the evolution equation (20), and from the constraint equation (23). If one were to take (curlh) = 0 along LRS curves, as was done in [8], then Eqs. (43)-(45) form a closed system, and the initial data specified in Section 2 is sufficient to determine H, H and E at all future times (up to singularities). If (curlh) 0, however, then this is no longer true. The value of (curlh) must be determined at future times, in order to evolve the system of evolution equations (43)-(45). This can be done by either numerically solving the full set of Einstein equations, or by performing a Taylor series expansion of (curlh) at the initial hypersurface [6]. In this paper we opt for the latter of these two methods, and in the remainder of this section we determine the coefficients of such an expansion using the compact notation from Section 3. In order to perform such an expansion, we first note that the time derivative of any spatial unit vector can be written as ė a 1 = e 1b u b u a Ω 3 e 2 a + Ω 2 e 3 a, (48) where Ω α := 1 2 ɛαβγ e βaė γa is the angular velocity of the spatial unit vectors, with respect to a set that are Fermi propagated along u a. If we choose our flow so that it is geodesic ( u a = 0), and take our unit vector n a to belong to a set of basis vectors that are Fermi-propagated (Ω α = 0), then we have ṅ a = 0 at all points along the evolution of all of our LRS curves. Letting (curl H ab ) (n) stand for the nth covariant derivative of curl H ab along u a, this means, in particular, that (curlh) (n) = n a n a (curl H ab ) (n), (49) for any number of time derivatives. We can therefore Taylor expand curl H ab, and simply contract the result with n twice, in order to find the series expansion of (curlh). 5.1 First Order The Taylor series expansion we wish to investigate is given by curl H ab = n=0 (curl H ab ) (n) t=0 n! t n, (50)

17 The Magnetic Part of the Weyl Tensor, and the Expansion of Discrete Universes 17 where (curl H ab ) (n) should be understood to be evaluated on the initial hypersurface. The time parameter, t, is the proper time that would be measured by an observer following the integral curves of u a. The n = 0 term in Eq. (50) is zero, as H ab must vanish at all points in the initial hypersurface. In fact, we can immediately see that every term that corresponds to an even value of n must vanish from this equation. This follows directly from the requirement of time-reversal symmetry. It now remains to evaluate the odd values of n. At first order we get (curlh ab ) = curl(θh ab ) + 3curl(σ c a H b c ) curl(curl(e ab )) (51) +3H c a H b c 1 3 Θcurl(H ab) σ e c ɛ cd(a D e H b) d, where we used the commutation relation between the curl operator and time derivatives (Eq. (A18) from [20]): (curls ab ) = curlṡab 1 3 ΘcurlS ab σ e c ɛ cd(a D e S b) d + 3H c a S b c, (52) where S ab is any projected symmetric tracefree tensor, S ab = S ab. The initial conditions in Eqs. (46) and (47) can then be seen to imply (curlh ab ) t=0 = 0, (53) at all points on the initial hypersurface. This means that the leading-order term in Eq. (50) cannot occur at any lower value than n = Third Order In order to find the third-order term in the Taylor series expansion, from Eq. (50), we must differentiate each term in Eq.(51) twice. For the first term we get (curl(θh ab )) = (ΘcurlH ab + ɛ cd(a H b) d D c Θ), (54) where we have again used the commutation rule from Eq. (52), as well as the following identity (Eq. (A16) from [20]): curl(fs ab ) = f curl(s ab ) + ɛ cd(a S b) d D c f, (55) where f is any scalar function. Applying the Leibniz rule, and plugging in the initial conditions, we then get (curl(θh ab )) t=0 = 0. For the second term in Eq.(51) we get [ 3(curl(σ c a H c b )) = 3 curl(σ c a H c b ) 1 3 Θcurl(σ c ah c b ) (56) σ e c ɛ c d (a D e σ g b) H d g + 3H c aσ d b H c d ] where we twice applied the identity from Eq. (52), and where we have also used vertical bars around indices to exclude them from symmetrization operations.

18 18 Timothy Clifton et al. The initial conditions stated in Eq. (46) then imply that (curl(σ c a H b c )) t=0 = 0, as all terms contain quantities that vanish at t = 0. The second time derivative of the fourth, fifth and sixth terms in Eq.(51) can also be seen to vanish, using similar logic. It is therefore only the third term on the right-hand side of Eq. (51) that can be non-zero, which means (curl(h ab )) t=0 = (curl curl(e ab )). Commuting the curl operator with the time derivatives, using Eq. (52) and the initial conditions from Eqs. (46) and (47), then gives (curl curl(e ab )) t=0 = curl curlëab curl(ɛ cd(a D e (E b) c E e d )). (57) This can be simplified further by using the following identity (Eq. (A4) from [20]): curl(s 2 ) ab = ɛ cd(a D e (S b) c S e d ), (58) where (S 2 ) ab := Sa c S bc. We then have, after using Eqs. (57)-(58) and the evolution equations (18)-(19), that (curl curl(e ab )) t=0 = 4curl curl(e 2 ) ab. This is the only term that is non-zero, after twice differentiating the right-hand side of Eq. (51). The final result, for the third derivative of curl H ab on the initial hypersurface, can therefore be written as (curlh ab ) t=0 = 4curl curl ( E 2) ab. (59) This is the first non-zero coefficient in the Taylor series from Eq. (50). Explicitly evaluating (curlh ab ), in terms of functions that appear in the metric, is likely to result in a very long expression, for all but the most trivial geometries. It is therefore remarkable that the same quantity can be written down in such a simple way in Eq. (59). This is a result of the compact notation that we have used, which hides an enourmous amount of underlying complexity. 5.3 Fifth Order The third-order coefficient, found in the previous section, is expected to be the leading-order term in the Taylor series expansion from Eq. (50), as it is the first one that does not vanish. In order to determine the regime in which the Taylor series provides a good approximation to the full geometry, however, it is necessary to find the next-to-leading order term (when the leading-order and next-to-leading-order terms become the same size, then we expect the series expansion to break down). We therefore need to calculate the fifth-order term in the series expansion from Eq. (50). The fifth-order term will be given by performing four differentiations on each term in Eq.(51). We will do this by following the same procedure used in the previous section, where simplifications were obtained using the identities in Eqs. (52), (55) and (58), and the initial conditions in Eqs. (46) and (47). Each of the six terms in Eq. (51) will be considered separately, and then summed to give out final result.

19 The Magnetic Part of the Weyl Tensor, and the Expansion of Discrete Universes First Term. To find the fourth derivative of the first term in Eq. (51), it is useful to first determine the second derivative without applying the initial conditions. This is given by (curl(θh ab )) = ΘcurlH ab 2 ΘcurlḢab Θ ΘcurlH ab + Θσ e c ɛ cd(a D e H b) d 3 ΘH c a H b c ΘcurlḦab Θ2 curlḣab + Θσ e c ɛ cd(a D e Ḣ b) d 3ΘH c a Ḣ b c Θ ΘcurlH ab Θ2 curlḣab 1 9 Θ3 curlh ab 1 3 Θ2 σ e c ɛ cd(a D e H b) d + Θ 2 H c a H b c + Θσ e c ɛ cd(a D e H b) d +Θσ e c ɛ cd(a (D e Ḣ b) d 1 3 ΘDe H b) d σ ed D d H b) d + 2H ec ɛ a e b)h d e ) +Θ σ e c ɛ cd(a D e H b) d 3 ΘH c a H b c 6ΘH c a Ḣ b c, (60) where we first expanded curl(θh ab ) using Eq. (55), and then applied the commutation relation from Eq. (52) twice. It can be seen from this equation that, if we differentiate twice more, then each term will contain at least one factor of a quantity that vanishes on the initial hypersurface. We therefore have curl(θh ab ) t=0 = 0, (61) so that the first term of Eq. (51) does not contribute to the fifth derivative of curl H ab Second Term. Applying Eq. (52), it can be seen that the second derivative of the second term in Eq. (51) is given by 3[curl(σ c a H b c )] = [3curl((σ c a H b c ) ) Θcurl(σ c a H b c ) 3σ e g ɛ gd(a D e (σ c b H d c ) + 9H d a σ c b H d c ].(62) This allows us to see, by inspection, that it is only the first term in this equation that will survive two further differentiations. After commuting these derivatives with the curl operator, and disregarding terms that vanish on the initial hypersurface, we then arrive at 3[curl(σ c a H b c )] t=0 = 3curl((σ c a H b c ) ). The only part of (σ c a H b c ) that does not vanish in the initial data is given by σ c a (H b c ). Using the evolution equations for σ ab and H ab, and the result of the previous section, in Eq. (59), then gives 3[curl(σ c a H b c )] t=0 = 48curl[E c a curl(e 2 ) b c ]. (63) This is the simplest expression we can find for this term.

20 20 Timothy Clifton et al Third Term. This is the most lengthy term to evaluate. We find, by repeated application of the commutation relation (52), that [curl(curle ab )] = curl((curle ab ) ) Θcurl((curlE ab) ) + σ e c ɛ cd(a D e ((curle b) d ) ) 3H c a (curle b c ) (Θcurl(curlE ab)) +(σ e c ɛ cd(a D e (curle b) d )) 3(H c a curle b c ). (64) The complete evaluation of the third term requires two further derivatives, and we can see that the only terms that survive this operation are given by [curl(curle ab )] t=0 = curl((curle ab ) ) + 6 σ e c ɛ cd(a D e ((curle b) d ) ), (65) where we have made use of the initial conditions. In order to evaluate this term we note that, after some manipulation, we can write (curle ab ) t=0 = 8curl(E 2 E ab ) + 30curl(E 3 ) ab + 4curl curl curl(e 2 ) ab 9E e c ɛ cd(a D e (E 2 ) b) d + 2(E 2 ) e c ɛ cd(a D e E b) d 9E e c ɛ cd(a D e (E 2 ) b) d + 12E c acurl(e 2 ) b c, (66) where we have again made repeated use of Eq. (52), as well as the following identity (Eq. (A12) from [20]): (D a S bc ) = D a Ṡ bc 1 3 ΘD as bc σa d D d S bc + 2Ha d ɛ de(b S e c). (67) We have also made the definition (S 3 ) ab := S ac S cd S db. Substituing Eq. (66) into Eq. (65), and performing some further manipulations, then gives [curl(curle ab )] = 8curl curl(e 2 E ab ) 30curl curl(e 3 ) ab + 18ɛ cd(a curl(d e (E e c (E 2 ) b) d ) 2ɛ cd(a curl((e 2 ) e c D e E b) d ) 12curl(E c acurl(e 2 ) b c ) 4curl curl curl curl(e 2 ) ab + 24ɛ cd(a D e (E e c curl(e 2 ) b) d ). (68) This is the lengthiest of the six terms being evaluated Fourth and Fifth Terms. The fourth and fifth terms, from Eq. (51), can both be seen to give 3(H c a H b c ) t=0 = 0, (69) and 1 3 (Θcurl(H ab)) t=0 = 0. (70)

21 The Magnetic Part of the Weyl Tensor, and the Expansion of Discrete Universes 21 This is because H ab and Θ, together with both their first and second derivatives, must vanish on the initial hypersurface. Two further time derivatives would therefore be required before either of these terms become non-zero at t = Sixth Term. The final term evaluates to [σ e c ɛ cd(a D e H b) d ] t=0 = 16E e c ɛ cd(a D e [curl(e 2 ) b) d ], (71) where we have used the commutation rules from Eqs. (52) and (55), as well as the evolution equations (18), (19) and (20). Putting all of these results together leads us to our final expression for the fifth order term in our Taylor series expansion of curl H ab : (curl(h) ab ) = 8 curl curl ( E 2 ) ( E ab 30 curl curl (E 3 ) ( ) ) ab 60 curl E c a curl(e2 ) b c ( ( )) ( ) +18 curl ɛ cd(a D e (E 2 ) d b) E e c 2 curl (E 2 ) e c ɛ cd(a D e E d ( b) ) +40ɛ cd(a D e curl(e 2 ) d b) E e c 4 curl curl curl curl(e 2 ) ab. (72) This equation is considerably less compact than the result for the third-order derivative, presented in Eq. (59), but is still remarkably short, given the complexity it conceals. In what follows, we will use the expressions calculated in Eqs. (59) and (72) to evaluate the first two non-vanishing terms in the Taylor series from Eq. (50). This will allow us to not only determine the leading-order effect that H ab has on the expansion of LRS curves, but also to estimate when the series expansion approach breaks down. We will perform this analysis for all three classes of LRS curves, and in all six of the lattices that were described in Section 2. 6 The Effect of Curl(H) on Cosmology The effect that H ab has on the expansion of space can be estimated by considering its consequences on the scale factors, a and a, defined implicitly by H = ȧ a and H = ȧ a. (73) The function a is then the scale factor along our LRS curves, and a is the scale factor in all perpendicular directions. Both are a function of proper time, along the integral curves of u, as well as position in space.

22 22 Timothy Clifton et al. Following the method used in [6], and extending it to higher order, we can use Eqs. (43)-(45) to write the correction to the scale factors, from including the curl H ab term in Eq. (45), as and a = a (curlh) t=0 t 6 (74) 6! + a [ 25 (curlh) E (curlh) ] t=0 t 8 + O(t 10 ) 8! a = a (curlh) t=0 t 6 (75) 2 6! a [ (curlh) E (curlh) ] t=0 t 8 + O(t 10 ), 2 8! where all quantities on the right-hand side of these equations should be taken to be evaluated at t = 0. These expressions result from simultaneously Taylor expanding a, a and (curl H) about the initial hypersurface, and substituting the results into Eqs. (43)-(45). The value of E is then given by noting that on a time-symmetric hypersurface, and in vacuum, the electric part of the Weyl tensor is equal to the Ricci tensor of the 3-space, E ab = (3) R ab. The quantities involving derivatives of curl H ab are given, in terms of E ab and its derivatives, by Eqs. (59) and (72). We therefore have all the information required to evaluate all of the terms present in Eqs. (75) and (76). The LRS curves that we will consider in this section are those displayed in Fig.1. The red curve in this figure extends halfway along a cell edge, from the centre of an edge to a vertex. The green curve connects the horizon of the mass at the centre with the centre of a cell face, and the blue curve connects the horizon and a cell vertex. These curves are depicted in Fig. 1 for the special case of a cubic cell, which is the basic constituent of the 8-cell. However, similar curves also exist for the case of tetrahedral, octahedral and dodecahedral cells. We will not present specific diagrams of these cells, but instead rely on the reader to visualize the corresponding curves in each of these cases. The proper length of each of our LRS curves, in a hypersurface of constant proper time, is then given by the integral L(t) = χ2 χ 1 a (t, χ)dχ, (76) where χ is the coordinate from Eq. (4), and where we have rotated the configuration so that the LRS curve under consideration is at constant θ and φ. In what follows, we will present the intial values of the functions E, as well as the derivatives (curl H) and (curl H), at every point along the three LRS curves depicted in Fig. 1. We will then calculate the proper length of each of these curves, as a function of proper time along the integral curves of u, until the Taylor expansion given in Eq. (50) breaks down. In calculating these results, we have substituted the expansion from Eq. (50) directly into the evolution equations (43)-(45).